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## G = C13×C8⋊C22order 416 = 25·13

### Direct product of C13 and C8⋊C22

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C4 — C13×C8⋊C22
 Chief series C1 — C2 — C4 — C52 — D4×C13 — C13×D8 — C13×C8⋊C22
 Lower central C1 — C2 — C4 — C13×C8⋊C22
 Upper central C1 — C26 — C2×C52 — C13×C8⋊C22

Generators and relations for C13×C8⋊C22
G = < a,b,c,d | a13=b8=c2=d2=1, ab=ba, ac=ca, ad=da, cbc=b3, dbd=b5, cd=dc >

Subgroups: 116 in 68 conjugacy classes, 40 normal (24 characteristic)
C1, C2, C2 [×4], C4 [×2], C4, C22, C22 [×5], C8 [×2], C2×C4, C2×C4, D4, D4 [×2], D4 [×2], Q8, C23, C13, M4(2), D8 [×2], SD16 [×2], C2×D4, C4○D4, C26, C26 [×4], C8⋊C22, C52 [×2], C52, C2×C26, C2×C26 [×5], C104 [×2], C2×C52, C2×C52, D4×C13, D4×C13 [×2], D4×C13 [×2], Q8×C13, C22×C26, C13×M4(2), C13×D8 [×2], C13×SD16 [×2], D4×C26, C13×C4○D4, C13×C8⋊C22
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, C13, C2×D4, C26 [×7], C8⋊C22, C2×C26 [×7], D4×C13 [×2], C22×C26, D4×C26, C13×C8⋊C22

Smallest permutation representation of C13×C8⋊C22
On 104 points
Generators in S104
(1 2 3 4 5 6 7 8 9 10 11 12 13)(14 15 16 17 18 19 20 21 22 23 24 25 26)(27 28 29 30 31 32 33 34 35 36 37 38 39)(40 41 42 43 44 45 46 47 48 49 50 51 52)(53 54 55 56 57 58 59 60 61 62 63 64 65)(66 67 68 69 70 71 72 73 74 75 76 77 78)(79 80 81 82 83 84 85 86 87 88 89 90 91)(92 93 94 95 96 97 98 99 100 101 102 103 104)
(1 60 80 102 30 16 74 47)(2 61 81 103 31 17 75 48)(3 62 82 104 32 18 76 49)(4 63 83 92 33 19 77 50)(5 64 84 93 34 20 78 51)(6 65 85 94 35 21 66 52)(7 53 86 95 36 22 67 40)(8 54 87 96 37 23 68 41)(9 55 88 97 38 24 69 42)(10 56 89 98 39 25 70 43)(11 57 90 99 27 26 71 44)(12 58 91 100 28 14 72 45)(13 59 79 101 29 15 73 46)
(14 45)(15 46)(16 47)(17 48)(18 49)(19 50)(20 51)(21 52)(22 40)(23 41)(24 42)(25 43)(26 44)(53 95)(54 96)(55 97)(56 98)(57 99)(58 100)(59 101)(60 102)(61 103)(62 104)(63 92)(64 93)(65 94)(66 85)(67 86)(68 87)(69 88)(70 89)(71 90)(72 91)(73 79)(74 80)(75 81)(76 82)(77 83)(78 84)
(14 58)(15 59)(16 60)(17 61)(18 62)(19 63)(20 64)(21 65)(22 53)(23 54)(24 55)(25 56)(26 57)(40 95)(41 96)(42 97)(43 98)(44 99)(45 100)(46 101)(47 102)(48 103)(49 104)(50 92)(51 93)(52 94)

G:=sub<Sym(104)| (1,2,3,4,5,6,7,8,9,10,11,12,13)(14,15,16,17,18,19,20,21,22,23,24,25,26)(27,28,29,30,31,32,33,34,35,36,37,38,39)(40,41,42,43,44,45,46,47,48,49,50,51,52)(53,54,55,56,57,58,59,60,61,62,63,64,65)(66,67,68,69,70,71,72,73,74,75,76,77,78)(79,80,81,82,83,84,85,86,87,88,89,90,91)(92,93,94,95,96,97,98,99,100,101,102,103,104), (1,60,80,102,30,16,74,47)(2,61,81,103,31,17,75,48)(3,62,82,104,32,18,76,49)(4,63,83,92,33,19,77,50)(5,64,84,93,34,20,78,51)(6,65,85,94,35,21,66,52)(7,53,86,95,36,22,67,40)(8,54,87,96,37,23,68,41)(9,55,88,97,38,24,69,42)(10,56,89,98,39,25,70,43)(11,57,90,99,27,26,71,44)(12,58,91,100,28,14,72,45)(13,59,79,101,29,15,73,46), (14,45)(15,46)(16,47)(17,48)(18,49)(19,50)(20,51)(21,52)(22,40)(23,41)(24,42)(25,43)(26,44)(53,95)(54,96)(55,97)(56,98)(57,99)(58,100)(59,101)(60,102)(61,103)(62,104)(63,92)(64,93)(65,94)(66,85)(67,86)(68,87)(69,88)(70,89)(71,90)(72,91)(73,79)(74,80)(75,81)(76,82)(77,83)(78,84), (14,58)(15,59)(16,60)(17,61)(18,62)(19,63)(20,64)(21,65)(22,53)(23,54)(24,55)(25,56)(26,57)(40,95)(41,96)(42,97)(43,98)(44,99)(45,100)(46,101)(47,102)(48,103)(49,104)(50,92)(51,93)(52,94)>;

G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13)(14,15,16,17,18,19,20,21,22,23,24,25,26)(27,28,29,30,31,32,33,34,35,36,37,38,39)(40,41,42,43,44,45,46,47,48,49,50,51,52)(53,54,55,56,57,58,59,60,61,62,63,64,65)(66,67,68,69,70,71,72,73,74,75,76,77,78)(79,80,81,82,83,84,85,86,87,88,89,90,91)(92,93,94,95,96,97,98,99,100,101,102,103,104), (1,60,80,102,30,16,74,47)(2,61,81,103,31,17,75,48)(3,62,82,104,32,18,76,49)(4,63,83,92,33,19,77,50)(5,64,84,93,34,20,78,51)(6,65,85,94,35,21,66,52)(7,53,86,95,36,22,67,40)(8,54,87,96,37,23,68,41)(9,55,88,97,38,24,69,42)(10,56,89,98,39,25,70,43)(11,57,90,99,27,26,71,44)(12,58,91,100,28,14,72,45)(13,59,79,101,29,15,73,46), (14,45)(15,46)(16,47)(17,48)(18,49)(19,50)(20,51)(21,52)(22,40)(23,41)(24,42)(25,43)(26,44)(53,95)(54,96)(55,97)(56,98)(57,99)(58,100)(59,101)(60,102)(61,103)(62,104)(63,92)(64,93)(65,94)(66,85)(67,86)(68,87)(69,88)(70,89)(71,90)(72,91)(73,79)(74,80)(75,81)(76,82)(77,83)(78,84), (14,58)(15,59)(16,60)(17,61)(18,62)(19,63)(20,64)(21,65)(22,53)(23,54)(24,55)(25,56)(26,57)(40,95)(41,96)(42,97)(43,98)(44,99)(45,100)(46,101)(47,102)(48,103)(49,104)(50,92)(51,93)(52,94) );

G=PermutationGroup([(1,2,3,4,5,6,7,8,9,10,11,12,13),(14,15,16,17,18,19,20,21,22,23,24,25,26),(27,28,29,30,31,32,33,34,35,36,37,38,39),(40,41,42,43,44,45,46,47,48,49,50,51,52),(53,54,55,56,57,58,59,60,61,62,63,64,65),(66,67,68,69,70,71,72,73,74,75,76,77,78),(79,80,81,82,83,84,85,86,87,88,89,90,91),(92,93,94,95,96,97,98,99,100,101,102,103,104)], [(1,60,80,102,30,16,74,47),(2,61,81,103,31,17,75,48),(3,62,82,104,32,18,76,49),(4,63,83,92,33,19,77,50),(5,64,84,93,34,20,78,51),(6,65,85,94,35,21,66,52),(7,53,86,95,36,22,67,40),(8,54,87,96,37,23,68,41),(9,55,88,97,38,24,69,42),(10,56,89,98,39,25,70,43),(11,57,90,99,27,26,71,44),(12,58,91,100,28,14,72,45),(13,59,79,101,29,15,73,46)], [(14,45),(15,46),(16,47),(17,48),(18,49),(19,50),(20,51),(21,52),(22,40),(23,41),(24,42),(25,43),(26,44),(53,95),(54,96),(55,97),(56,98),(57,99),(58,100),(59,101),(60,102),(61,103),(62,104),(63,92),(64,93),(65,94),(66,85),(67,86),(68,87),(69,88),(70,89),(71,90),(72,91),(73,79),(74,80),(75,81),(76,82),(77,83),(78,84)], [(14,58),(15,59),(16,60),(17,61),(18,62),(19,63),(20,64),(21,65),(22,53),(23,54),(24,55),(25,56),(26,57),(40,95),(41,96),(42,97),(43,98),(44,99),(45,100),(46,101),(47,102),(48,103),(49,104),(50,92),(51,93),(52,94)])

143 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 8A 8B 13A ··· 13L 26A ··· 26L 26M ··· 26X 26Y ··· 26BH 52A ··· 52X 52Y ··· 52AJ 104A ··· 104X order 1 2 2 2 2 2 4 4 4 8 8 13 ··· 13 26 ··· 26 26 ··· 26 26 ··· 26 52 ··· 52 52 ··· 52 104 ··· 104 size 1 1 2 4 4 4 2 2 4 4 4 1 ··· 1 1 ··· 1 2 ··· 2 4 ··· 4 2 ··· 2 4 ··· 4 4 ··· 4

143 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 4 4 type + + + + + + + + + image C1 C2 C2 C2 C2 C2 C13 C26 C26 C26 C26 C26 D4 D4 D4×C13 D4×C13 C8⋊C22 C13×C8⋊C22 kernel C13×C8⋊C22 C13×M4(2) C13×D8 C13×SD16 D4×C26 C13×C4○D4 C8⋊C22 M4(2) D8 SD16 C2×D4 C4○D4 C52 C2×C26 C4 C22 C13 C1 # reps 1 1 2 2 1 1 12 12 24 24 12 12 1 1 12 12 1 12

Matrix representation of C13×C8⋊C22 in GL4(𝔽313) generated by

 48 0 0 0 0 48 0 0 0 0 48 0 0 0 0 48
,
 0 0 1 0 0 0 0 312 0 1 0 0 1 0 0 0
,
 1 0 0 0 0 312 0 0 0 0 0 312 0 0 312 0
,
 1 0 0 0 0 1 0 0 0 0 312 0 0 0 0 312
G:=sub<GL(4,GF(313))| [48,0,0,0,0,48,0,0,0,0,48,0,0,0,0,48],[0,0,0,1,0,0,1,0,1,0,0,0,0,312,0,0],[1,0,0,0,0,312,0,0,0,0,0,312,0,0,312,0],[1,0,0,0,0,1,0,0,0,0,312,0,0,0,0,312] >;

C13×C8⋊C22 in GAP, Magma, Sage, TeX

C_{13}\times C_8\rtimes C_2^2
% in TeX

G:=Group("C13xC8:C2^2");
// GroupNames label

G:=SmallGroup(416,197);
// by ID

G=gap.SmallGroup(416,197);
# by ID

G:=PCGroup([6,-2,-2,-2,-13,-2,-2,1273,3818,9364,4690,88]);
// Polycyclic

G:=Group<a,b,c,d|a^13=b^8=c^2=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^3,d*b*d=b^5,c*d=d*c>;
// generators/relations

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